Vector geometry and midpoint arguments · Higher
| English | 中文 | Pinyin |
|---|---|---|
| collinear/ˈkɒlɪnɪə/ | 共线 | gòng xiàn |
A triangle’s two side midpoints create a new segment. Vectors can prove that it is parallel to the third side and exactly half its length.
- A triangle’s two side midpoints create a new segment. Vectors can prove that it is parallel to the third side and exactly half its length.
- This lesson studies collinear 共线: Lying on one straight line.
Choose the mathematical structure
- Add/subtract components and multiply a vector by a scalar. Position vectors locate points from one origin; displacement AB=b-a joins two points. A scalar multiple gives parallel directions; to establish collinearity, also connect the displacements to a common point. Midpoints average position vectors. Give a chain of vector equalities with clear start and end points.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines collinear?
Lying on one straight line.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Let OA=a and OB=b, with M midpoint of OA and N midpoint of OB. Then OM=a/2 and ON=b/2, so MN=b/2-a/2=(b-a)/2=AB/2. Thus MN is parallel to AB and half as long. For a=(4,2),b=(2,6), M=(2,1),N=(1,3), AB=(-2,4) and MN=(-1,2), verifying the general result. In parallelogram OACB with OC=a+b, the midpoint of OC is (a+b)/2; the midpoint of AB is the same, so the diagonals bisect each other. If AP=3AB/2, P lies on line AB beyond B; if AP=AB/2, P is its midpoint. A parallel vector at another location alone does not prove three specified points collinear.
Vector geometry and midpoint arguments
Add/subtract components and multiply a vector by a scalar
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
For a=(4,2),b=(2,6), find the x-component of AB.
b-a gives 2-4=-2.
Test a tempting shortcut
- Keep AB=b-a, not a-b. Parallelism alone does not locate a line. A numerical example can check the algebra but cannot replace the general midpoint proof.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
If two vectors are parallel, any three points used in their diagrams must be collinear. This claim is false. Explain which definition or assumption it violates.
Find the y-component of midpoint M of OA.
Half of a_y=2 is 1.
If two vectors are parallel, any three points used in their diagrams must be collinear.
Keep AB=b-a, not a-b. Parallelism alone does not locate a line. A numerical example can check the algebra but cannot replace the general midpoint proof.
Interpret a new situation
- AQA G25 Higher uses vectors for geometric arguments/proofs, while G24 and the basic G25 operations also belong to Foundation. This proof lesson avoids scalar products and spatial line equations.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the y-component of MN for the two midpoint vectors.
Half of b_y-a_y=(6-2)/2=2.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Higher · 3.4. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
Lying on one straight line. Choose the relationship, show the method, check its assumptions and interpret the result.